Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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46 views

Is there any “realistic” metric of type $\mathrm{(A)dS}_2 \times \mathcal{T}_2$ in General Relativity?

After studying the Bertotti-Robinson metric, which describes a $\mathrm{AdS}_2 \times \mathcal{S}_2$ universe, I was wondering about other kind of closed topologies with holes, like $\mathrm{(A)dS}_2 \...
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Two particles at different points on a curved manifold

Stupid question, but why two particles at different points on a curved manifold do not have any well-defined notion of relative velocity? For instance n cosmology the light from distant galaxies is ...
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49 views

Non-linear Perturbations of Minkowski Spacetime

I am reading some of the following paper on the bounded $L^2$ conjecture in general relativity where it mentions non-linear perturbations of the Minkowski metric in the context of quasilinear wave ...
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Proof of equivalence of variational principle and Euler-Lagrange equations on a manifold [closed]

Let M be some manifold, and TM the tangent bundle. Let $\gamma : [a,b] \to M$ be a smooth curve on M defined on an interval on $\mathbb{R}$. Let $J$ be another interval in $\mathbb{R}$ containing 0. A ...
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Does $\mathcal{M} = AdS_2 \otimes S_2$ makes any sense as a manifold?

I'm not a topologist or a group theorist and I need a clarification about some notations. Consider the Bertotti-Robinson metric in General Relativity (relativity students should study this metric, by ...
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78 views

Orthochronous indefinite orthogonal group $O^+(m, n)$ form a group

My question is based on Qmechanic's answer here which proves that $O^+(m, 1)$ forms a group -- that if two Lorentz transformations have positive time-time co-ordinate, so does their product. The key ...
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56 views

Globally constant vector field in a curved spacetime

Is it possible to define a globally constant vector field in a curved spacetime, that is a vector field for which the covariant derivative vanishes along every world line? The vector field $V^{\mu}=0$ ...
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28 views

Harper-Hofstadter model in symmetric gauge

If I have l a square lattice, with the total flux = $\pi$, I can work in the symmetric gauge, which will have my vector potential be $A = \frac{\pi}{2}(-y,x)$. In a tight-binding model with Peirels ...
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1answer
87 views

Cylindrical universe cosmology in general relativity

Is there a compact cylindrical universe solution to the Einstein equation, with space homogeneity, without using "artificial" periodic boundaries? I'm expecting a metric of the following shape: \...
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43 views

Electromagnetism on 3 torus

We all know Maxwell equations in 3+1 spacetime, where the "space" is $\mathbb{R}^3$ and time is $\mathbb{R}$. Moreover, it is easy to construct (using differential forms) the corresponding theory in a,...
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Understanding Skein relations

I'm just studying some concepts in the topology, but I cannot figure out totally how Skein relations work. in the following I've one example: which tries to relate Reidmeister II to the Skein ...
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56 views

How to know which topological invariant is in play?

I'm currently working on the Haldane model where I've worked through the math to find that when the condition $$ \frac{M}{t_2} = 3 \sqrt{3} sin (\phi) $$ is satisfied the gap closes, meaning there ...
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Does a topological charge always need to be an integer

Does a topological charge always need to be an integer, I see many papers where people talk about non-integer topological charges due to boundary conditions. According to the formula for the ...
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1answer
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What if Newton's bucket had been a sphere?

My question involves a modification of Newton's bucket experiment. If a sphere filled (say) one-third or one-half with water is rotated very very fast, will the water eventually spread out across and ...
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What are the transformation equations for deviatoric planes?

What are the transformation equations $$r=r(x,y,c),$$ $$\phi=\phi(x,y,c)$$ for a Matsuoka-Nakai deviatoric plane? Variables $x,y$ are Cartesian coordinates, $r,\phi$ are "polar" coordinates, $c$ ...
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Does the Standard Model have texture defects?

In the standard classification of topological defects, in a theory with vacuum manifold $\mathcal{M}$, $\pi_0(\mathcal{M})$ corresponds to domain walls, $\pi_1(\mathcal{M})$ corresponds to strings/...
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37 views

In what sense do bifurcations concern change in quality?

I've heard such vague statements several times and also read: Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family. (From ...
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What makes a topological insulator topological?

I understand that a topological insulator is one with an insulating bulk and conducting surface but I don't understand why or how the topological part comes into it. All of the resources I've found ...
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Does CMB rule out that the universe is infinite?

If the universe were infinite, the energy of the big bang would have been long dissipated, and very little or nothing would hit us. Does the fact that CMB still comes roughly the same from every ...
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1answer
102 views

Why is $\rm{Conf}(\mathbb{R}^{1,1}) = \rm{Diff}(S^1) \times \rm{Diff}(S^1)$ and not $ \rm{Diff}(\mathbb{R}) \times \rm{Diff}(\mathbb{R})$?

The Minkowski metric for $\mathbb{R}^{1,1}$ is $$ ds^2 = dt^2 - dx^2 = du dv $$ for coordinates $$ u = t + x \hspace{1cm} v = t - x $$ If you do any coordinate transformation that acts independently ...
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Spacetime manifolds of 1+1 d systems for writing TQFT partition functions

Are the spacetime manifolds of two unentangled systems disconnected? This arises in the context of thinking of an operator whose expectation value we wish to take by writing this quantity in terms ...
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1answer
109 views

What's the meaning of “inequivalent quantizations”?

The notion "inequivalent quantizations" is regularly used when topological terms are discussed. From what I've gathered so far, "inequivalent quantizations" means that there are different quantum ...
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What happens to large gauge transformations in gauges different from the temporal gauge?

There are already several questions regarding the meaning and definition of large gauge transformations. Discussions of large gauge transformations typically only happen in the context of the ...
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56 views

Wave function as a section of a complex line bundle to do QM in polar coordinates

If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum ...
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Is it mathematically impossible to incorporate the space curvature into the equations of motion and gravity? [duplicate]

Obviously, I haven't studied GR, I know no more than common knowledge. However, I'm wondering, is it impossible to develop a mathematical model based on flat space, in which the new equations of ...
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Lefschetz and Witten indices$.$

I couldn't help but notice a formal similarity between the Lefschetz index $$ \mathrm{ind}(f)=\sum_k (-1)^k\operatorname{tr}(f_*|H_k) $$ and the Witten index $$ Z=\operatorname{tr}((-1)^Fe^{-\beta H}) ...
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Is there a definition for a *geometric entropy*?

In statistical mechanics, entropy of a system is usually defined as a measure of the system's micro-state randomness, or as an averaged "surprise" of its micro-state: \begin{equation}\tag{1} S_{\text{...
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53 views

Understanding Anyonic Exchange

In the book of "Introduction To Topological Quantum Computation" by Jiannis K. Pachos, in chapter 5, it tries to explain anyonic exchange. In the following, the $m$ and $e$ quasi-particles are ...
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Rigourous formalism of Hamiltonian mechanics on Manifolds

I'm looking for books / articles that provide rigorous formulations of Hamiltonian mechanics on Manifolds. I found the book "Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds" [1]...
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Reference for topology for topological insulators

In the field of topological insulators What topological space do they talk of? Looking for some resources that sheds light on the topology part of topological insulator
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2-sheeted Riemann surface with 2 branch cuts and Torus

A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that ...
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Particle on a circle with magnetic flux$.$

I am trying to understand the model studied in 1905.09315 §2, to wit, a $0+1$ dimensional theory with target space $\mathbb S^1$ with non-trivial magnetic flux: $$ \mathcal L=\frac12m\dot q^2-\frac{i}{...
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Pulling apart the atoms of a topological insulator

Consider a topological insulator. In order to destroy a topological phase, the band gap of the bulk system should close at some point (passing thru a conducting state), but if the atoms that make up ...
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Fluxes on a finite group $G$

So I've been studying about topological quantum computation and I have a few questions I haven't been able to solve. The first one is why fluxes take values on a finite group $G$? Does it have to do ...
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137 views

Rigorous procedure of gluing together two spacetimes

There seems to exist a procedure of "gluing two spacetimes together". In particular I've seem this mentioned in the context of gravitational collapse. The examples I've seem where that of gluing ...
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What is the shape of the universe? [duplicate]

If it's flat then how a volume can be flat? And I've read that it's actually not flat .....it's a "Poincaré dodecahedral space". Any suggestions for books posts or articles are highly appreciated.
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Does the Einstein field equation uniquely determines the topology of spacetime? [duplicate]

I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was ...
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String topology in string theory

How do string topology, string field theory and topological strings interact? Does anybody see a global picture? By string topology I mean the TQFT based on the homology of the space of loops ...
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Can the shape of our Universe be a Mobius strip? [duplicate]

The Friedmann Equations describe three possibilities for the shape of our using General Relativity, I read in a book that the shape of our Universe is a 3-sphere such that in any direction if you ...
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Critical points of vector field with zeros in the magnitude

I am studying a vector field which has critical points (sources, sinks, saddle points and centers). The magnitude of the vector field goes to zero smoothly in these points, however. Contrast that to ...
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86 views

Berry phase: Spin in a magnetic field parameter space manifold

Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce ...
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1answer
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Topological theta-term as a background electric/magnetic field?

The topological $\theta$-term in the Schwinger model (1+1-dimensional QED) can be interpreted as a background electric field, as explained in Chapter 7.1.2 of Tong's lecture notes. The same holds true ...
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1answer
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Theta-dependence of massive Schwinger model

I've read in Coleman's paper on the massive Schwinger model (and in other papers on the same topic, like this one) that the model's Hamiltonian contains a topological $\theta$-term. However, if I ...
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1answer
143 views

Black holes in p-adic gravity/ultra-metric metric field? [closed]

As a radically different to beyond standard general relativity, at least from the type of geometry it deals with: Consider p-adic gravity and/or general relativity defined on certain ultra-metric ...
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What’s the topology of critical region?

Duhem said the aim of physics is natural classification. I think topology and geometry are a wonderful way to link analogous parts among different phenomena. Thus we can classify and predict facts. ...
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200 views

“Hidden” theta-term in Hamiltonian formulation of Yang-Mills theory

I've read in David Tong's lecture notes on gauge theory that the Hamiltonian of Yang-Mills theory does not depend on the angular parameter $\theta$, because it can be absorbed in the electric field: $...
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Symmetry operators of a Bloch Hamiltonian

Consider a lattice with a 3 atom basis, e.g. the Lieb lattice, and some completely arbitrary on-site energies and hopping energies and phases between the different atoms. In momentum space we can ...
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1answer
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Change in area on a 3-sphere bounded by a trajectory due to a differential change in trajectory

I have a 3 dimensional spherical topology, and I draw a curve onto the sphere labelled by $\vec{n}(\vec{r},t)$. The area bounded by the curve is termed the "Wess Zumino Action" (Hence my motivation to ...
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Hausdorff property in Minkowski spacetime

In the 4-dimensional Minkowski spacetime, for a given point $x = (x^0,x^1,x^2,x^3)$, its timelike future/past set is defined as, $$ I^{\pm}(x) = \{y =(y^0,...,y^3) \in \mathbb{R}^4 : \eta_{\mu \nu}(y-...
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For the torus to rotate 180 degrees around the East-West symmetry axis, what happens?

(Suppose to ignore the deformed friction and torus when rotating)